Multi-objective optimization of injection molding energy consumption and product quality using a combined approach of RSM and BP neural network

Abstract

With the global environmental challenges and the urgent energy-saving needs of the polymer material processing industry, an innovative multi-objective optimization method of energy consumption and product quality for injection molding process parameters was proposed. The influence mechanism of seven key process parameters on the synergistic optimization of energy consumption and quality was systematically investigated by constructing a dedicated experimental platform. The hybrid assignment strategy integrating grey correlation analysis and entropy weight method was adopted to quantitatively evaluate the three major indexes of injection molding energy consumption, product quality and tensile strength. The mathematical model was established by using the response surface method (RSM) and BP neural network combined with genetic algorithm respectively, and the optimization was computed by using the Design-Expert and MATLAB software, so as to realize the multi-objective optimization of product quality and energy consumption. The results showed that both optimization methods obtained good results, in which the response surface method optimizes the injection molding process with energy consumption decreased by 6.73%, product tensile strength increased by 7.06%, and the comprehensive optimization rate was 13.81%. The BP neural network-genetic algorithm optimized the injection molding process with energy consumption decreased by 9.32%, product tensile strength enhanced by 4.81%, and the comprehensive optimization rate was 14.15%.

Introduction

The injection molding stands out as the most widely utilized and fundamental method for plastic molding. ¹ In the injection molding industry, the energy consumed in molding process occupies a significant portion of the overall production process. Energy saving in the injection molding process essentially means that the power system of the equipment outputs less energy to get the same quality of product. The injection molding equipment is expensive and the price of an injection molding equipment ranges from tens of thousands to several million. Compared with this, the energy consumed in the injection molding process is negligible, but with the gradual increase in the proportion of the number of plastic products in the modern production and life, even if a single piece of plastic products production of the electricity required to reduce the not 1%, but this way of approaching the promotion of the method. Even if the electricity required for the production of a single plastic product is reduced by less than 1%, if this approach and method are extended to the whole industry, the value of energy saved in the whole plastics processing industry will be quite objective. Some scholars specializing in the analysis of this issue, such as a plastic product production can save 1% of the energy consumption, then the annual savings of about 279.36 billion kilowatt-hours of electricity, equivalent to a medium-sized thermal power plant 582 days of power generation.^2,3

The injection molding machine consumes a large amount of energy during the production process. Elduque ⁴ measured the energy consumption data of 36 plastic parts produced by 12 different injection molding machines (including hydraulic, hybrid, and fully electric). The results show that the all-electric injection molding machine has the lowest specific energy consumption (SEC), saving 54.4% energy compared to traditional models. Therefore, reasonable optimization and selection of the structure and system of the injection molding machine can effectively achieve the goal of energy saving. The energy consumption of the power part accounts for the highest proportion of the power consumption of the whole machine. Liu ⁵ explored the energy efficiency of asynchronous motors, servo motors and hydraulic pumps under different working conditions, and then, compared and analyzed the energy consumption of injection molding machines driven by five different types of electro-hydraulic power units to get the lowest energy-consuming power unit in the injection molding process. The energy consumption of the hydraulic drive part of a traditional injection molding machine accounts for about 60%-70% of the energy consumption of the whole machine, so the renovation and upgrading of the hydraulic control system has long been one of the hotspots for energy saving in injection molding. The hydraulic pulsation injection molding machine invented by Prof. Jinping Qu^6–8 cleverly applies the vibration mechanism of polymers in the molding process, which improves product quality and reduces energy consumption. Jiao ⁹ proposed a new type of internal circulation two-plate injection molding machine closing mechanism for the characteristics of large energy consumption in the process of mold removal of precision fully hydraulic injection molding machine, and the experimental tests on its energy consumption showed that it can greatly improve the energy-saving characteristics of injection molding machine. Mianehrow ¹⁰ analyzed the energy consumption characteristics of hydraulic injection molding machines through experiments and found that output and cycle time are key process parameters that affect specific energy consumption (SEC), while machine parameters (such as hydraulic pump type, control system) mainly determine the peak power consumption. The study emphasizes that optimizing process design is more effective in reducing energy consumption than replacing new machines, and reveals the differences in energy consumption curves of different machines, providing a theoretical basis for energy-saving improvements in industry.

Although energy consumption can be effectively reduced by selecting and improving the structure of the injection machine in the injection molding process, such as drive and actuation, the cost of retrofitting and the difficulty of upgrading are self-evident, so it has become extremely important to explore the methods of reducing energy consumption on the basis of the existing machines from multiple perspectives. Most scholars focus on the multi-objective effects of process parameters on energy consumption and quality. Meekers et al ¹¹ experimentally revealed the dominant role of the cooling time (32% of the total energy consumption) and proposed that a parameter lowering strategy can significantly reduce the resource consumption while safeguarding the quality. Deng et al ¹² further elucidated the dual role of screw speed: the reliance on barrel heating at low speeds leads to energy loss, while high-speed shear heat enhances plasticizing efficiency, providing a thermodynamic basis for speed optimization. This finding echoes the study of extrusion energy consumption by Abeykoon ¹³ pointed out that the coupling effect between the rheological properties of the material and the screw speed leads to a decrease in the specific energy consumption with the increase in speed. In order to break through the high cost limitations of experimental optimization, Weissman et al ¹⁴ proposed a CAD model-based energy consumption prediction method to quantify energy consumption at the design stage through Moldflow analysis, which promotes the forward movement of energy consumption management. Wen ¹⁵ studied the calculation method and influencing factors of energy consumption in injection molding process, and constructed a high-precision energy consumption calculation model using MATLAB software, providing an effective method for optimizing injection molding process parameters and reducing energy consumption. Spiering ¹⁶ constructed an energy consumption knowledge base covering production lines and workshops, established specific energy consumption (SEC) as a key performance indicator (KPI), and provided a standardized framework for industry energy efficiency benchmarking.

In recent years, the integration of intelligent modeling and optimization algorithms has provided a new technological path to solve the problem of effective balance between quality, efficiency, and energy consumption that traditional single optimization methods find difficult to achieve. Yeh et al ¹⁷ taking a plastic screw as an example, combined the Taguchi method and response surface methodology (RSM) to conduct parameter optimization research aimed at energy consumption and multiple quality characteristics. They found that packing time and cooling time are key factors influencing energy consumption, and through parameter optimization, achieved a 29%–33% reduction in energy consumption, while also verifying that RSM outperforms the Taguchi method in handling interactions and nonlinear relationships. Wang et al ¹⁸ aiming to reduce the energy consumption of tower mills in mineral processing, proposed a modeling and optimization method based on a genetic algorithm-optimized BP neural network. By optimizing process parameters, they effectively reduced grinding energy consumption, demonstrating the feasibility and accuracy of this intelligent optimization approach in lowering energy consumption through parameter adjustment. Paşcoschi et al ¹⁹ investigated energy consumption in the injection molding of fruit containers and proposed a novel application strategy for artificial intelligence algorithms, combining an unsupervised autoencoder with the K-Means algorithm to analyze production data and identify key factors affecting injection molding energy consumption. Their results indicate that the application of AI algorithms can effectively reduce energy consumption in plastic molding processes. To further enhance modeling accuracy and optimization capability, researchers have progressively introduced hybrid strategies combining artificial neural networks (ANN) and evolutionary algorithms. Oktora et al ²⁰ proposed an intelligent optimization method integrating ANN and differential evolution (DE). Using data obtained from full factorial experimental designs, they constructed a high-precision prediction model capable of simultaneously predicting part weight and energy consumption. By employing the DE algorithm, they significantly reduced energy consumption with only a slight sacrifice in part weight. Jou et al ²¹ developed a hybrid framework based on a back-propagation neural network (BPNN) and particle swarm optimization (PSO). By optimizing the network structure using the Taguchi method, they achieved reduced energy consumption and minimized material waste while ensuring product quality. In the context of global multi-objective optimization, Guo et al ²² proposed a comprehensive optimization approach integrating neural networks, the NSGA-II multi-objective optimization algorithm, and a fuzzy decision-making method based on the Critic method. By establishing a PSO-BP neural network prediction model, they accurately mapped the nonlinear relationships between process parameters and energy consumption, quality, and warpage. This approach significantly reduced energy consumption while ensuring product lightweighting. El Ghadoui et al ²³ combined a back-propagation neural network with a genetic algorithm to systematically optimize multiple objectives, including dimensional accuracy, weight, and energy consumption. Ultimately, they significantly reduced raw material consumption, cycle time, and specific energy consumption while maintaining product quality. Furthermore, addressing specific molding defects such as warpage deformation, Nitnara, Hong, and Fu et al^24–26 employed artificial intelligence and genetic algorithms combined with finite element simulations to optimize process parameters in injection molding. Their approaches effectively predicted and controlled warpage deformation, significantly improving product quality in injection molding.

Although the existing research has made remarkable progress, most of the research only stays in the theoretical or simulation stage, without experimental research. Moreover, most studies only consider energy saving in injection molding, without balancing the relationship between product quality and energy consumption. The field urgently needs to be based on the theory of energy consumption and experiments, research and analysis of key factors affecting the energy consumption of injection molding, as well as to solve the problem of product energy saving at the same time to achieve the best quality and other issues. Both the molding quality and molding energy consumption are closely related to the injection molding process parameters. This paper relies on the existing experimental equipment, to explore the injection molding process parameters on the energy consumption of the mechanism, under the premise of ensuring product quality, to seek for the lowest energy consumption production of the optimal process parameters.

Both molding quality and molding energy consumption are closely related to injection molding process parameters. Therefore, the mechanism of the influence of injection molding process parameters on energy consumption were explored in the paper, the optimal process parameters for the lowest energy consumption production were studied. Figure 1 Shows the specific experimental process. Utilize existing experimental equipment to establish an experimental platform, and investigate the relationship between seven process parameters and injection molding energy consumption through response surface experimental design. Additionally, incorporate quality indicators (tensile strength, weight) as co-evaluation metrics. Based on grey relational analysis and the entropy weight method, the three indicators are fitted into a single metric. Process parameter optimization analysis is conducted using the response surface method and BP neural network genetic algorithm. The optimization results of both methods are compared to analyze their advantages and disadvantages, ultimately identifying the process parameter combination that yields the lowest energy consumption for injection-molded tensile specimens.

Figure 1.

Technology road map.

Material and Methods

Material

The sample material used in this work was polypropylene in the form of pellets and with a trade mark 5090T (MFI = 15 g/10 min), supplied by the Formosa petrochemical Corp, Taiwan, China.

Equipment and Instruments

Micro-injection molding machine: the experimental work was carried out on an injection molding machine of type BOY XS, having a maximum injection pressure 2298 bar, with screw diameter for plastication 14 mm and maximum weight of the product 6.1 g.

Mold: The multi-spline injection mold constructed from two parts (tensile specimen and impact specimen). The mold cavity thickness is 1 mm. The cavity pressure and temperature are measured in the mold cavity by the quartz sensor for mold cavity pressure type Kistler 6190CA, which has a front of 4.0 mm diameter. Data output from the amplifier is collected using a Kistler 5865 Como injection system. Computer is used to record the output reading of the acquisition system through an interface cart by the help of lab. view program.

Figure 2 Presents the dimensions of both the specimen and the gate, illustrating the critical geometric parameters employed in this study.

Figure 2.

Dimension of specimen and gate.

Mold temperature controller: The mold temperature controller (model TP6ZE) was adopted.

Chiller (model ML-CA03) was adopted.

Electronic balance (model CP214) was adopted. The accuracy is 0.1 mg.

LMG Energy consumption meter (model LMG600) was adopted.

Mechanical properties (model KL-WS-30S) were adopted.

Test and Characterization

Figure 3 Shows the complete process of measuring energy consumption, weight, and tensile strength. Five specimen experiments were performed under the same number, and the average value of each group of experiments was taken, and relative error was calculated. The LMG energy consumption meter was connected to the corresponding channel of the BOY-XS injection molding machine to measure the energy consumption value of the injection molding process. The collection frequency of the energy consumption meter was set to 50 ms. The energy consumption measurement and data processing method was as follows: After heating the screw and hydraulic oil of the injection molding machine to the set temperature, test the mold 50 times first, and conduct the experiment after the injection molding machine runs stably. When starting the experiment, click the “start” button on the energy consumption meter operation panel to measure the power. Each set of experimental parameters were tested continuously for 5 times, and the average value was taken as the experimental result. The energy consumption meter records the power consumption of the injection molding machine every 50 ms. According to W = $\int \mathrm{p}$ dt, which is the integration of power over time, the energy consumption value of the machine during that time period can be obtained. The spline obtained from the experiment was measured for weight using an OHAUS tester and mechanical properties were tested using a universal tensile tester.

Figure 3.

Complete process work for energy consumption, weight and tensile strength determination.

Design of Response Surface Experiments

The energy consumption (X), weight (Y) and tensile strength (Z) were selected as the response variables for the experiment, and mold temperature (A), injection pressure (B), holding pressure (C), holding time (D), screw temperature (E), V/P switching position (F) and screw rotational speed (G) as the factor variables.

Table 1 Shows the factor levels selected for response surface experiments. The level values were selected uniformly according to the recommended parameter ranges and the actual production experience to establish the seven-factor, three-level experimental design.

Table 1.

Response surface experiment factor level table.

Level	A/°C	B/bar	C/bar	D/S	E/°C	F/mm	G/%
−1	40	80	80	3	200	2	18
0	50	90	90	5	220	6	24
1	60	100	100	7	240	10	30

Table 2 Shows the results of response surface experiments. According to the process parameters designed in the above experiment, energy consumption and quality analysis of the injection molding process were carried out using experimental equipment BOY-XS micro injection molding machine, energy consumption monitoring equipment LMG-600, universal tensile tester, and weight tester.

Table 2.

Experimental design of response surface.

Number	A/°C	B/bar	C/bar	D/S	E/°C	F/mm	G/%	X/J	Y/g	Z/MPa
1	60	90	90	5	220	2	18	4280.81	0.5458	38.8700
2	50	100	100	5	220	2	24	4299.89	0.5453	34.904
3	50	90	100	3	220	6	30	3588.16	0.551	37.5840
4	40	90	90	5	220	2	30	4153.83	0.5407	38.108
5	60	100	90	7	220	6	24	4539.7	0.5456	35.4310
6	40	80	90	3	220	6	24	3555.07	0.5473	37.6490
7	50	80	80	5	220	2	24	4248.49	0.5418	37.3105
8	50	100	80	5	220	2	24	4146.88	0.5423	36.8720
9	40	90	80	5	200	6	24	3741.32	0.5401	33.3770
10	50	80	90	5	240	6	30	3971.85	0.5468	37.3075
11	50	80	80	5	220	10	24	3930.09	0.5418	34.7775
12	50	100	100	5	220	10	24	4142.59	0.5545	40.0285
13	60	90	80	5	240	6	24	3967.17	0.5444	37.6195
14	50	100	90	5	240	6	30	3918.44	0.5460	36.7630
15	50	90	80	3	220	6	18	3684.24	0.5420	38.0135
16	50	90	90	3	200	10	24	3616.81	0.5469	32.1355
17	40	90	90	5	220	2	18	3930.70	0.5472	39.4760
18	40	90	80	5	240	6	24	3965.32	0.5443	38.5990
19	40	90	100	5	240	6	24	4037.79	0.5541	36.6310
20	50	90	100	7	220	6	18	4597.47	0.5490	37.4550
21	40	90	90	5	220	10	30	3903.46	0.5475	34.8595
22	50	80	100	5	220	2	24	4210.25	0.5497	38.9170
23	60	90	100	5	200	6	24	4028.70	0.5510	33.5330
24	40	100	90	7	220	6	24	4323.78	0.5467	37.9930
25	60	90	90	5	220	10	30	4023.21	0.5485	34.7920
26	50	90	90	5	220	6	24	3979.93	0.5450	37.3665
27	60	90	90	5	220	10	18	4145.69	0.5404	36.1510
28	50	80	100	5	220	10	24	4114.05	0.5541	38.0400
29	50	90	90	3	240	2	24	3696.66	0.5469	38.3050
30	50	90	90	5	220	6	24	4036.33	0.5461	37.4395
31	50	90	90	3	200	2	24	3868.82	0.5450	35.9835
32	40	80	90	7	220	6	24	4294.72	0.5466	38.1370
33	50	90	90	5	220	6	24	4049.25	0.5435	37.6280
34	50	100	90	5	240	6	18	4036.56	0.5479	36.2955
35	60	90	90	5	220	2	30	4333.93	0.5444	34.4480
36	50	90	100	3	220	6	18	3729.3	0.5532	37.0475
37	50	80	90	5	200	6	18	4048.43	0.5425	34.3185
38	50	90	100	7	220	6	30	4443.69	0.5492	37.7990
39	50	100	80	5	220	10	24	4088.42	0.5433	35.6655
40	50	90	90	7	240	2	24	4650.38	0.5438	38.1930
41	50	80	90	5	200	6	30	3885.19	0.5461	34.9920
42	50	90	90	5	220	6	24	4047.02	0.5461	38.0930
43	50	80	90	5	240	6	18	4008.55	0.5479	36.8955
44	60	90	80	5	200	6	24	4016.13	0.5416	35.6840
45	60	80	90	3	220	6	24	3715.29	0.5463	36.1220
46	50	100	90	5	200	6	30	3847.53	0.5455	35.4780
47	50	90	90	7	240	10	24	4272.94	0.5454	35.8775
48	60	100	90	3	220	6	24	3577.12	0.5474	36.0840
49	50	90	90	5	220	6	24	3906.2	0.5463	37.2635
50	40	90	90	5	220	10	18	4030.29	0.5458	35.4275
51	50	90	80	3	220	6	30	3581.51	0.5416	38.1255
52	50	90	90	5	220	6	24	4161.72	0.5459	37.4080
53	60	80	90	7	220	6	24	4437.89	0.5444	37.3840
54	60	90	100	5	240	6	24	4220	0.5545	37.0485
55	50	100	90	5	200	6	18	4168.87	0.5473	33.5975
56	50	90	80	7	220	6	18	4454.14	0.5391	38.8520
57	50	90	90	7	200	10	24	4336.95	0.5435	32.9795
58	50	90	80	7	220	6	30	4347.2	0.5410	39.0525
59	50	90	90	3	240	10	24	3558.06	0.5488	34.8040
60	40	100	90	3	220	6	24	3536.45	0.5460	38.0810
61	40	90	100	5	200	6	24	4068.22	0.5410	33.8040
62	50	90	90	7	200	2	24	4524.86	0.5437	34.3600

Signal-to-noise ratio and dimensionless

The signal-to-noise ratio is a measure of the importance of experimental factors on the results and an important basis for judging the robustness of output characteristics. It is usually converted into dB values. According to different usage scenarios, the signal-to-noise ratio can be divided into Nominal-the-Best, Larger-the-Better and Smaller-the-Better. For the total energy consumption of injection molding (X), it is hoped that its value is as small as possible, so the Smaller-the-Better is used, which can be calculated according to the Smaller-the-Better signal-to-noise ratio calculation equation (1). For the weight (Y), it is hoped that its value is as close as possible to the target value, so the Nominal-the-Best is used, which can be calculated according to the Nominal-the-Best signal-to-noise ratio calculation equation (2). For the tensile strength (Z), it is desired that its value is as large as possible, so the Larger-the-Better is used, which can be calculated according to the Larger-the-Better signal-to-noise ratio in equation (3).

${\eta }_s=\frac{s}{n}=-10\mathit{lg}\left(\frac{1}{n}\sum _{i=1}^n{y}_i^2\right)$

(1)

${\eta }_n=\frac{s}{n}=-10\mathit{lg}\left[\frac{1}{n-1}{\displaystyle \sum }_{i=1}^n{\left(y_i-\frac{1}{n}{\displaystyle \sum }_{i=1}^n{y}_i\right)}^2\right]$

(2)

${\eta }_l=\frac{s}{n}=-10\mathit{lg}\left(\frac{1}{n}{\displaystyle \sum }_{i=1}^n\frac{1}{{y_i}^2}\right)$

(3)

In the equation: $n$ -- sample size; $y_i$ -- comparison sequence.

Since the three investigated metric have entirely different dimensions, the calculation of the Signal-to-Noise Ratio cannot eliminate the discrepancies caused by unit variations. Therefore, it is necessary to further perform dimensionless normalization on the SNR data, also known as normalization processing. Equation (4), (5) and (6) are three different dimensionless calculation methods.

$\text{Nominal}-\text{the}-\text{Best}:X_i^{\mathit{※}}\left(k\right)=1-\frac{\left|X_i^{\left(0\right)}\left(k\right)-X^{\left(0\right)}\right|}{\mathit{max}\hspace{0.17em}{X}_i^{\left(0\right)}-X^{\left(0\right)}}$

(4)

$\text{Larger}-\text{the}-\text{Better}:X_i^{\mathit{※}}\left(k\right)=\frac{\left|X_i^{\left(0\right)}\left(k\right)-\mathit{min}\hspace{0.17em}{X}_i^{\left(0\right)}\left(k\right)\right|}{\mathit{max}\hspace{0.17em}{X}_i^{\left(0\right)}-\mathit{min}\hspace{0.17em}{X}_i^{\left(0\right)}\left(k\right)}$

(5)

$\text{Smaller}-\text{the}-\text{Better}:X_i^{\mathit{※}}\left(k\right)=\frac{\left|{maxX}_i^{\left(0\right)}\left(k\right)-X_i^{\left(0\right)}\left(k\right)\right|}{\mathit{max}\hspace{0.17em}{X}_i^{\left(0\right)}-\mathit{min}\hspace{0.17em}{X}_i^{\left(0\right)}\left(k\right)}$

(6)

In the equation: $X_i^{\mathit{※}}\left(k\right)$ -- Dimensionless results; $X_i^{\left(0\right)}\left(k\right)$ --the signal-to-noise ratio; $X^{\left(0\right)}$ --Nominal-the-Best target value; ${maxX}_i^{\left(0\right)}\left(k\right)$ --Sequence Maximum; $\mathit{min}\hspace{0.17em}{X}_i^{\left(0\right)}\left(k\right)$ --Sequence Minimum.

The further calculation and organization of the results after dimensionless can get the gray correlation coefficient, which characterizes the relationship between the dimensionless data and the ideal data, and the calculation equation is shown in equation (7):

$\gamma \left(X_0^{\mathit{※}}\left(k\right),X_i^{\mathit{※}}\left(k\right)\right)=\frac{{\mathrm{\Delta }}_{\mathit{min}}+\xi {\mathrm{\Delta }}_{\mathit{max}}}{\left|X_0^{\mathit{※}}\left(k\right)-X_i^{\mathit{※}}\left(k\right)\right|+\xi {\mathrm{\Delta }}_{\mathit{max}}}$

(7)

In the equation: $X_0^{\mathit{※}}$ --Ideal value of data,Larger-the-Better Median value of 1,Smaller-the-Better Median value of 0; $\xi$ --Distinguishing coefficient, $\xi \in \left[0,1\right]$ ,typically set as $\xi =0.5$ ; ${\mathrm{\Delta }}_{\mathit{max}}$ --Maximum value after dimensionless,1; ${\mathrm{\Delta }}_{\mathit{min}}$ -- Minimum value after dimensionless,0.

Entropy Weighting Method for Calculating Weighting Coefficients

The three quality metric are calculated to obtain three grey correlation coefficients, which need to be integrated into a comprehensive evaluation index for subsequent optimization. Entropy weight method is a way of objective assignment method, mainly based on the entropy value of each metric to determine the weight coefficient. The principle is that the smaller the degree of change of the metric, the larger the entropy value, the less information and the weight coefficient corresponding to it is also smaller. The method and steps for determining the weight coefficients by entropy weighting method are as follows:

(a) The experimental results are constructed into a decision matrix, which is randomly regularized, as shown in equation (8):

${\lambda }_{ij}=\frac{x_{ij}}{\sqrt{{\displaystyle \sum }_{i=1}^m{x}_{ij}^2}}$

(8)

In the equation: ${\lambda }_{ij}$ -- represents the regularized outcome of the j-metric in the ith experimental trial,i = 1,…,n, j = 1,…,m, n = 3, m = 62; $x_{ij}$ -- represents the outcome of the jth metric measured in the ith experimental trial.

(b) Calculate the proportion of the jth metric relative to the total sum of the data in the ith experimental trial, as shown in equation (9):

$p_{ij}=\frac{{\lambda }_{ij}}{{\displaystyle \sum }_{i=1}^m{\lambda }_{ij}}$

(9)

(c) Calculate the entropy of each metric, as shown in equation (10):

$e_j=-\frac{{\displaystyle \sum }_{i=1}^m{p}_{ij}\hspace{0.17em}\mathit{ln}\left(p_{ij}\right)}{\mathit{ln}\left(m\right)}$

(10)

(d) Calculating information redundancy,as shown in equation (11):

$d_j=1-e_j$

(11)

(e) Calculation of weights, as shown in equation (12):

$w_j=\frac{d_j}{{\displaystyle \sum }_{i=1}^n{d}_j}\times 100\%$

(12)

Table 3 Shows the weight results of each indicator calculated according to the entropy weight method steps. Since the weight of the study object of the experiment is small and its variation is small, so its weight is small.

Table 3.

Metric weights.

Metric	Energy consumption X (%)	Weights Y (%)	Tensile strength Z (%)
Weights	66.98	0.41	32.61

Gray Correlation Calculation

According to equation (13), the gray correlation of each experiment was calculated.

$\varphi \left(X_0^{\mathit{※}},X_i^{\mathit{※}}\right)=\underset{i=1}{\overset{n}{{\displaystyle \Sigma }}}{w}_j·\gamma \left(X_0^{\mathit{※}}\left(k\right),X_0^{\mathit{※}}\left(k\right)\right)$

(13)

According to the above steps, the three experimental metrics signal-to-noise ratio, dimensionless value, gray correlation coefficient, and gray correlation are computed sequentially, the since the gray correlation directly characterizes the strengths and weaknesses of the three metric, this study refers to it as the composite score Q. The larger its value, the closer it represents to the desired value.

Table 4 Shows the results with composite scoring as the experimental objective.

Table 4.

Experimental composite scores.

Experiment number	Composite scores Q	Experiment number	Composite scores Q	Experiment number	Composite scores Q
1	53.98	22	55.56	43	53.86
2	42.31	23	46.95	44	50.81
3	81.43	24	49.44	45	66.32
4	53.46	25	48.98	46	57.21
5	39.41	26	56.19	47	44.67
6	85.61	27	48.06	48	78.76
7	48.66	28	54.23	49	58.77
8	49.78	29	74.07	50	49.99
9	59.89	30	54.63	51	84.05
10	56.31	31	57.19	52	51.18
11	52.27	32	50.53	53	45.47
12	63.82	33	54.70	54	48.51
13	61.53	34	51.48	55	43.17
14	57.07	35	40.90	56	50.73
15	73.89	36	67.53	57	38.86
16	68.6	37	47.43	58	53.48
17	66.93	38	46.69	59	78.63
18	61.15	39	48.59	60	89.81
19	52.21	40	45.34	61	46.10
20	43.44	41	54.63	62	37.73
21	53.53	42	56.40

Results and Discussion

Response Surface Method Optimization

The regression analysis of process parameters and experimental results were carried out by using Design-Export software. The items were fitted by the least-squares method and the fitted model was significant with a P-value of less than 0.0001 by ANOVA. The composite score itself was fitted to the three indicators by a certain response relationship to the process parameters, so the response relationship of the process parameters to the score was also obvious.

Figure 4 Shows the normal distribution of residuals obtained from the comprehensive evaluation. The closer the residual value distribution is to the straight line, the better the fitting relationship. The graph shows that the normal probability distribution of residuals is more in line with the straight line, and each point is distributed on the straight line or in the vicinity of both sides of the line, indicating a good fitting effect.

Figure 4.

Plot of normal distribution of residuals.

Figure 5 Shows the distribution of the residual of the comprehensive score and the predicted values of the regression equation. The looser the distribution of points indicates that the model has good applicability. The loose point distribution in the figure shows no obvious regularity, indicating that the fitting model has good applicability.

Figure 5.

Distribution of residuals of predicted values.

Figure 6 Shows the correspondence between actual values and predicted values during the prediction validation of the fitting model. The straight line represents the actual value, and the closer the point is to the straight line, the more accurate the prediction and the better the model. Each point of the predicted value in the figure is basically distributed on the straight line or near both sides of the straight line, which indicates that the experimental and predicted values of the comprehensive score are basically consistent, further increasing the credibility of the response surface model.

Figure 6.

Correspondence between actual values and predicted values.

The optimization of the constructed response surface model was carried out based on the Design-Export response surface method. The optimal value of the composite score is 94.75, which was the best value of the composite score. The corresponding process parameters were: mold temperature 40°C, injection pressure 100 bar, holding pressure 94.85 bar, holding time 3 s, screw temperature 228.88°C, V/P switching position 9.35 mm, and screw speed 29.96%.

BP neural networks and genetic algorithm optimization

BP Neural Network

According to the content of the previous study, the relationship between injection molding quality, energy consumption and process parameters is not only linear and interactive influence. Fitting the relationship between them through the least squares method will inevitably have significant errors. According to the previous research, the relationship between injection quality, injection energy consumption, and process parameters is not just linear and interactive. Fitting the relationship between them through the least squares method will inevitably have significant errors. BP neural network regards the relationship between process parameters and comprehensive indicators as a “network” and trains it through a sufficient number of sample experiments. BP neural network continuously adjusts its adaptability to reduce errors, and can ultimately obtain a sufficiently accurate function relationship for prediction.

According to the 62 groups of data in Table 5., the process parameters corresponding to each group of data are used as the input values of BP neural network, and the comprehensive score is used as the output value, the first 52 groups of data are selected as the training samples, and the remaining 10 groups of data are used as the test samples to check the degree of goodness of the training network. In order to make the network converge faster in iterations, the original dataset is normalized using the mapmin-max function and the network is trained using MATLAB software.

Table 5.

Experimental range of values for process parameters.

Mold temperature/°C	Injection pressure/bar	Holding pressure/bar	Holding time/s	Screw temperature/°C	V/P switching position/mm	Screw speed/%
40-60	80-100	80-100	3-7	200-240	2-10	18-30

The overall score was found to be 98.21 after 155 iterations, corresponding to the input values of mold temperature 40°C, injection pressure 90.37 bar, holding pressure 82.9 bar, holding time 3s, screw temperature 240°C, V/P switching position 5.55 mm, and screw speed 18.90%.

Figure 7 Shows the BP neural network structure, in which the seven process parameters are used as the seven nodes of the input layer, and the corresponding composite score values are used as the nodes of the output layer, in which the determination of the implied layer nodes is more complicated, and is generally calculated according to the empirical equation (14) and (15). But the empirical equation only gives a range of values for the number of nodes in the implied layer, it is necessary to modify the number of nodes of the implied layer during training in order to obtain an accurate training model. It is necessary to continuously modify the number of hidden layer nodes during training, starting from the minimum value to obtain an accurate training model, until a model with sufficiently small mean square error is output. After training, the number of hidden layer nodes is determined to be 10, and finally, a BP neural network with 7-10-1 nodes in each layer is formed.

$N=\sqrt{N_{input}+N_{output}}+\alpha$

(14)

$N={\mathit{log}}_2\left(input\right)+\alpha$

(15)

In the equation: $N$ -- Number of nodes in the implicit layer; $N_{input}$ —Number of nodes in the input layer; $N_{output}$ —Number of nodes in the output layer; $\alpha$ — constant 1∼10.

Figure 7.

BP neural network structure.

The tansig function was chosen as the transfer function for the implicit layer and Pureline was chosen as the transfer function for the output layer. The number of nodes in the input layer was 7, the number of nodes in the implicit layer was 10, the number of nodes in the output layer was 1, the maximum number of training times was set to 1000, the minimum error of the training objective was 0.0001, and the learning efficiency was set to 0.01.

Figure 8 Shows the true value and the predicted value error comparison chart. MATLAB software was used to implement BP neural network training, the remaining 10 sets of data compared with the network predicted value. It can be seen that the trained BP neural network had a superior prediction function, the predicted comprehensive scoring system are in the vicinity of the target value, the error was relatively small, so the training of the resulting network performance was excellent. To fully demonstrate the accuracy of the trained BP neural network model, the model was evaluated using two evaluation metrics: coefficient of determination $R^2$ and root mean square error RMSE, according to formulas (16) and (17), the $R^2$ of the prediction model is 0.985 and RMSE is 1.82, indicating a high correlation between the predicted and actual values. The fitting effect of the prediction model is very good, with high prediction accuracy and generalization ability.

$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}$

(16)

$R^2=1-\frac{\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}{\sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}$

(17)

Figure 8.

BP neural network prediction validation plot.

Genetic Algorithm Global Optimization

The trained BP neural network model was used as its fitness function, with the genetic population size was set to 100, crossover probability was set 0.4, variance rate was set 0.2, and number of iterations was set 100. The range of values of input seven variables was set based on the previously described.

Table 5 Shows the ranges of the experimental parameters.

Validation of Results

The response surface method and BP neural network-GA were used to optimize the injection molding process parameters, and the results were verified by using the optimized process parameters for actual production in the previous section. Among them, the optimized process parameters of response surface method were shown in Figure 9 And the optimized energy consumption of injection molding by BP neural network-GA method was shown in Figure 10.

Figure 9.

Response surface methodology optimization of injection molding energy consumption.

Figure 10.

Neural network-GA optimization of injection molding energy consumption.

Further test specimen quality indicators: weight and tensile strength. The experimental group with the highest comprehensive score during the whole experiment in the previous section was the 60th group, with a comprehensive score of 89.81, and its injection energy consumption was 3536.45 J, weight was 0.546 g, and tensile strength was 38.08 MPa, and it will be used as a control group to compare the optimization effect, and the optimization result was placed in Table 6. It can be seen that both methods had better optimization results, in which the combined optimization rate of the response surface method was 13.81%, while the combined optimization rate of the BP neural network-GA method was 14.15%, the former optimization results in higher tensile strength, and the latter optimization results in lower energy consumption of injection molding. Since the weight of the object in this study is small, the optimization of weight is not obvious, and both methods control the weight around the average weight.

Table 6.

Optimization results.

Experimental group	Energy consumption	Optimization rate	Weights	Optimization rate	Tensile strength	Optimization rate
Group 60	3536.45		0.5460		38.08
Response surface	3298.4	6.73%	0.5461	0.02%	40.77	7.06%
BP-GA	3206.39	9.32%	0.5459	0.02%	39.91	4.81%

Conclusion

The analysis on the energy consumption of injection molding was carried out by combining theory and experiment in this paper. The effect of different process parameters on injection molding energy consumption and product quality was explored by using response surface experiments. The multi-objective optimization of the injection molding process was studied by using response surface methodology and BP neural network genetic algorithm. The main conclusions obtained are as follows:

The injection energy consumption, weight and tensile strength were taken as the optimization indexes, and seven process parameters were taken as the optimization objects, combining the grey theory and entropy weighting method to carry out multi-objective optimization for the three indexes. Based on the Design-Export response surface method, the constructed response surface model was optimized, and the optimal value was 94.75, the comprehensive optimization rate is 13.81%, corresponding to the following process parameters: mold temperature 40°C, injection pressure 100 bar, holding pressure 94.85 bar, holding time 3s, screw temperature 228.88°C, V/P switching position 9.35 mm, and screw rotation speed 29.96%. Based on MATLAB BP neural network and genetic algorithm for global optimization, after 155 iterations, the optimal comprehensive score of 98.21 was found, with a comprehensive optimization rate of 14.15%, corresponding to the input values of mold temperature 40°C, injection pressure 90.37 bar, holding pressure 82.9 bar, holding time 3s, screw temperature 240°C, V/P switching position 5.55 mm, and screw speed 18.90%. Among them, the BP neural network method has the highest optimization rate for injection molding energy consumption, and the response surface method has the highest optimization rate for tensile strength. If a company has higher quality requirements for products in actual production, the response surface method can be considered for optimization while meeting the demand for injection molding energy consumption. However, considering all factors, the BP neural network method has the highest optimization rate.